\[
  \mathrm{id}_{A} \xrightarrow{\ \beta\ } gf , \qquad
  fg \xrightarrow{\ \alpha\ } \mathrm{id}_{B}
  \tag{1.1}
\]

\[
  \varphi = fg : B \longrightarrow B
  \tag{1.2}
\]

\[
  \varphi \xrightarrow{\ \alpha\ } \mathrm{id}_{B} , \qquad
  \varphi \xrightarrow{\ \lambda\ } \varphi^{2}
  \tag{1.3}
\]

\[
  \lambda : fg \longrightarrow fgfg , \qquad \lambda = f * \beta * g
  \tag{1.4}
\]

\[
  K = K(\varphi, \alpha, \lambda)
  \tag{1.7}
\]

\[
  (X, u) \qquad X \in \mathrm{Ob}\, B, \quad u : X \longrightarrow \varphi(X)
  \tag{1.8}
\]

\[
  h : A \longrightarrow K(\varphi, \alpha, \lambda)
  \tag{1.11}
\]

\[
  \varphi : B \longrightarrow B
  \tag{2.1}
\]

\[
  \varphi \xrightarrow{\ \alpha\ } \mathrm{id}_{B} , \qquad
  \varphi \xrightarrow{\ \lambda\ } \varphi^{2}
  \tag{2.2}
\]

\[
  A = K(\varphi, \alpha, \lambda)
  \tag{2.3}
\]

\[
  A \xrightarrow{\ f\ } B
  \tag{2.4}
\]

\[
  g : B \longrightarrow A
  \tag{2.5}
\]

\[
  g(X) = \bigl( \varphi(X),\ \varphi(X) \xrightarrow{\ \lambda(X)\ } \varphi^{2}(X) \bigr)
  \tag{2.6}
\]

\[
  fg = \varphi
  \tag{2.7}
\]

\[
  fg \xrightarrow{\ \alpha\ } \mathrm{id}
  \tag{2.8}
\]

\[
  \alpha' : \mathrm{Hom}_{A}(X, g(Y)) \longrightarrow \mathrm{Hom}_{B}(f(X), Y) ,
  \qquad w \mapsto \alpha \circ f(w)
  \tag{2.9}
\]

\[
  \beta : \mathrm{id}_{K} \longrightarrow gf
  \tag{2.10}
\]

\[
  \beta' : \mathrm{Hom}_{B}(f(X), Y) \longrightarrow \mathrm{Hom}_{A}(X, g(Y)) ,
  \qquad w \mapsto g(w) \circ \beta(X)
  \tag{2.11}
\]

\[
  B = \prod_{i \in I} B_{i}
  \tag{3.1}
\]

\[
  f_{i} : A \longrightarrow B_{i}
  \tag{3.2}
\]

\[
  g_{i} : B_{i} \longrightarrow A ,
  \tag{3.3}
\]

\[
  g\bigl( (X_{i})_{i \in I} \bigr) = \prod_{i \in I} g_{i}(X_{i})
  \tag{3.4}
\]

\[
  \mathrm{pr}_{j}\, \varphi\bigl( (X_{i}) \bigr)
  = f_{j}\Bigl( \prod_{i \in I} g_{i}(X_{i}) \Bigr) .
  \tag{3.5}
\]

\[
  \varphi_{ji} \overset{\text{déf}}{=} f_{j} g_{i} : B_{i} \longrightarrow B_{j}
  \tag{3.6}
\]

\[
  \mathrm{pr}_{j}\, \varphi\bigl( (X_{i}) \bigr)
  = \prod_{i \in I} \varphi_{ji}(X_{i})
  \tag{3.6}
\]

\[
  \prod_{i} \varphi_{ki}(X_{i}) \xrightarrow{\ \bar{\alpha}_{k}\ } X_{k}
  \tag{3.7}
\]

\[
  \prod_{i} \varphi_{ki}(X_{i}) \xrightarrow{\ \bar{\lambda}_{k}\ }
  \prod_{i,j} \varphi_{kj} \varphi_{ji}(X_{i})
  \tag{3.8}
\]

\[
  \prod_{\alpha} \varphi_{k\alpha}(X_{\alpha})
  \xrightarrow{\ \bar{\lambda}_{k,j,i}\ } \varphi_{kj} \varphi_{ji}(X_{i})
\]

\[
  \left\{
  \begin{array}{l}
    \bar{\alpha}_{k} = \alpha_{k}(X_{k}) \circ \mathrm{pr}_{k} \\[2pt]
    \bar{\lambda}_{k,j,i} = \lambda_{kji}(X_{i}) \circ \mathrm{pr}_{i}
  \end{array}
  \right.
  \tag{3.9}
\]

\[
  \lambda_{kji} : \varphi_{ki} \longrightarrow \varphi_{kj} \circ \varphi_{ji}
  \tag{3.10}
\]

\[
  \mathrm{id}_{B_{i}} \longrightarrow \varphi_{ij} \circ \varphi_{ji}
  \qquad (i \neq j)
\]

\[
  B = B' \times B'' , \quad f = (f', f'') , \quad
  g(X', X'') = g'(X') \times g''(X'')
  \tag{3.12}
\]

\[
  \varphi'' : B' \longrightarrow B'' , \qquad
  \varphi' : B'' \longrightarrow B'
  \tag{3.13}
\]

\[
  \left\{
  \begin{array}{l}
    \mathrm{id}_{B'} \xrightarrow{\ \lambda'\ } \varphi' \varphi'' \\[2pt]
    \mathrm{id}_{B''} \xrightarrow{\ \lambda''\ } \varphi'' \varphi'
  \end{array}
  \right.
  \tag{3.14}
\]

\[
  \varphi(X', X'') = \bigl( X' \times \varphi'(X'') ,\ \varphi''(X') \times X'' \bigr)
\]

\[
  X' \xrightarrow{\ u'\ } \varphi'(X'') , \qquad
  X'' \xrightarrow{\ u''\ } \varphi''(X')
  \tag{3.15}
\]

\[
  (X', X'') \xrightarrow{\ u\ }
  \bigl( X' \times \varphi'(X'') ,\ X'' \times \varphi''(X') \bigr)
  \rightrightarrows \varphi^{2}(X', X'')
\]

\[
  \bigl( X', X'',\ X' \xrightarrow{\ u'\ } \varphi'(X'') ,\
  X'' \xrightarrow{\ u''\ } \varphi''(X') \bigr)
\]

\[
  \left\{
  \begin{array}{l}
    g'(X') = \bigl( X',\ \varphi''(X'),\
      X' \xrightarrow{\lambda'(X')} \varphi'\varphi''(X') ,\
      \varphi''(X') \xrightarrow{\mathrm{id}} \varphi''(X') \bigr) \\[3pt]
    g''(X'') = \bigl( \varphi'(X''),\ X'',\
      \varphi'(X'') \xrightarrow{\mathrm{id}} \varphi'(X'') ,\
      X'' \xrightarrow{\lambda''(X'')} \varphi''\varphi'(X'') \bigr)
  \end{array}
  \right.
  \tag{3.17}
\]

\[
  \left\{
  \begin{array}{l}
    \text{Image essentielle de } g' : u'' \text{ est un isomorphisme} \\[2pt]
    \text{Image essentielle de } g'' : u' \text{ est un isomorphisme}
  \end{array}
  \right.
  \tag{3.18}
\]

\[
  \begin{array}{ccl}
    X & = & \bigl( X', X'',\ X' \xrightarrow{u'} \varphi'(X'') ,\
      X'' \xrightarrow{u''} \varphi''(X') \bigr) \\[4pt]
    \downarrow \beta' & & \\[4pt]
    g' f'(X) & = & \bigl( X',\ \varphi''(X'),\
      X' \xrightarrow{\lambda'(X')} \varphi'\varphi''(X') ,\
      \varphi''(X') \xrightarrow{\mathrm{id}} \varphi''(X') \bigr)
  \end{array}
  \tag{3.20}
\]

\[
  \phi : \mathrm{id}_{C} = E^{0} \longrightarrow E , \qquad
  \lambda : E^{2} \longrightarrow E
\]

\[
  \phi' : E' \longrightarrow \mathrm{id}_{C'} = E'^{0} , \qquad
  \lambda' : E' \longrightarrow E'^{2}
\]

\[
  h_{B}(A) = \mathrm{Hom}(A, B) .
\]

\[
  \alpha : \mathrm{Hom}(B, B') \xrightarrow{\ \sim\ }
  \mathrm{Hom}_{C}(h_{B}, h_{B'})
\]

\[
  M(h) = h(U) ,
\]

\[
  \alpha' : \mathrm{Hom}(h, h') \xrightarrow{\ \sim\ }
  \mathrm{Hom}_{R^{\circ}}\bigl( M(h), M(h') \bigr)
\]

\[
  \beta : \mathrm{Hom}_{R^{\circ}}\bigl( M(h), M(h') \bigr)
  \longrightarrow \mathrm{Hom}(h, h')
\]

\[
  T''(M) = M \otimes_{R} U .
\]

\[
  \left\{
  \begin{array}{l}
    \mathrm{Hom}_{C}(U,\ M \otimes_{R} U) \simeq M \\[2pt]
    \text{si } R = \mathrm{Hom}_{C}(U, U) , \quad M \in \mathrm{Mod}_{R^{\circ}}
  \end{array}
  \right.
\]

\[
  \mathrm{Hom}_{C'}\bigl( S(A), X \bigr) \simeq
  \mathrm{Hom}_{C}\bigl( A, T(X) \bigr)
  \tag{1}
\]

\[
  \mathrm{Hom}_{C}(A, B) \longrightarrow
  \mathrm{Hom}_{C'}\bigl( S(A), S(B) \bigr) \simeq
  \mathrm{Hom}_{C}\bigl( A, TS(B) \bigr)
  \tag{2}
\]

\[
  \varphi(A) \in \mathrm{Hom}_{C}\bigl( A, TS(A) \bigr) , \quad \text{i.e.}
  \tag{3}
\]

\[
  \varphi : \mathrm{id}_{C} \longrightarrow TS
  \tag{3 bis}
\]

\[
  \mathrm{Hom}_{C'}\bigl( ST(Y), X \bigr) \simeq
  \mathrm{Hom}_{C}\bigl( T(Y), T(X) \bigr)
  \tag{4}
\]

\[
  \psi(X) \in \mathrm{Hom}_{C'}\bigl( ST(X), X \bigr) , \quad \text{i.e.}
  \tag{5}
\]

\[
  \psi : ST \longrightarrow \mathrm{id}_{C'}
  \tag{5 bis}
\]

\[
  S(A,B) : \mathrm{Hom}_{C}(A,B) \longrightarrow
  \mathrm{Hom}_{C'}\bigl( S(A), S(B) \bigr)
\]

\[
  0 \longrightarrow A' \longrightarrow A \longrightarrow A'' \longrightarrow 0
  \tag{1}
\]

\[
  0 \longrightarrow S(A') \longrightarrow S(A) \longrightarrow S(A'')
  \longrightarrow 0
  \tag{1 bis}
\]

\[
  0 \longrightarrow \mathrm{Hom}(A'', M) \longrightarrow \mathrm{Hom}(A, M)
  \longrightarrow \mathrm{Hom}(A', M) \longrightarrow 0
  \tag{2}
\]

\[
  0 \longrightarrow \mathrm{Hom}\bigl( S(A''), X \bigr) \longrightarrow
  \mathrm{Hom}\bigl( S(A), X \bigr) \longrightarrow
  \mathrm{Hom}\bigl( S(A'), X \bigr) \longrightarrow 0
  \tag{2 bis}
\]

\[
  \Sigma(A) : \quad 0 \longrightarrow A \longrightarrow TS(A)
  \longrightarrow Z(A) \longrightarrow 0
  \tag{3}
\]

\[
  0 \longrightarrow S(A) \longrightarrow STS(A) \longrightarrow SZ(X)
  \longrightarrow 0
\]

\[
  0 \longrightarrow \mathrm{Hom}\bigl( Z(A), M \bigr) \longrightarrow
  \mathrm{Hom}\bigl( TS(A), M \bigr) \longrightarrow
  \mathrm{Hom}(A, M) \longrightarrow 0
\]

\[
  S\bigl( \mathrm{Ker}\, v / \mathrm{Im}\, u \bigr) =
  \mathrm{Ker}\, S(v) / \mathrm{Im}\, S(u)
\]

\[
  C^{1}(A) = C^{0}(A) / \mathrm{Im}\, A , \qquad
  C^{i}(A) = C^{0}\Bigl( C^{i-1}(A) / \mathrm{Im}\bigl( C^{i-2}(A) \bigr) \Bigr)
  \quad \text{pour } i \geqslant 2
\]

\[
    0 \longrightarrow \Gamma\bigl( C^{0}(A') \bigr) \longrightarrow
    \Gamma\bigl( C^{0}(A) \bigr) \longrightarrow
    \Gamma\bigl( C^{0}(A'') \bigr) \longrightarrow 0
  \]

\[
  0 \longrightarrow \mathrm{Hom}(\widetilde{A''}, \widetilde{N})
    \longrightarrow \mathrm{Hom}(\widetilde{A}, \widetilde{N})
    \longrightarrow \mathrm{Hom}(\widetilde{A'}, \widetilde{N})
    \longrightarrow 0 .
\]

\begin{equation*}
  S_X : \quad 0 \longrightarrow X \xrightarrow{\ i_X\ } T(X)
    \longrightarrow X^{*} \longrightarrow 0
  \tag{8}
\end{equation*}

\[
  S(S_X) : \quad 0 \longrightarrow \widetilde{X} \longrightarrow
  \widetilde{T(X)} \longrightarrow \widetilde{X^{*}} \longrightarrow 0 ,
\]

\[
  0 \longrightarrow \widetilde{A} \longrightarrow \widetilde{X}^{0}
    \longrightarrow \widetilde{X}^{1} \longrightarrow \widetilde{X}^{2}
    \longrightarrow \cdots
\]

\[
  0 \longrightarrow \widetilde{Z}^{i}(X^{\bullet}) \longrightarrow
  \widetilde{X}^{i} \longrightarrow \widetilde{Z}^{i+1}(X^{\bullet})
  \longrightarrow 0 \qquad (i \geqslant 0).
\]

\[
  \bar{u}^{n+1} \;=\; t^{n}\, T(\bar{u}^{n})\, s'^{n+1} ,
\]

\[
  C^{n+1}(A) \;=\; T\bigl( C^{n}(A) / \mathrm{Im}\, C^{n-1}(A) \bigr)
  \qquad \text{(avec par convention } C^{-1}(M) = A\text{)} .
\]

\[
  C = C^{U} \longrightarrow \widetilde{C} = C^{V}
  \qquad \text{restriction des scalaires.}
\]

\[
  \mathrm{Hom}_{V}(\widetilde{A}, \widetilde{B})
  \;=\; \mathrm{Hom}_{V}(A, B)
  \;\xrightarrow{\ \sim\ }\;
  \struck{\mathrm{Hom}_{U}(U \otimes_{V} A, B)}\;
  \mathrm{Hom}_{U}\bigl( A, \mathrm{Hom}_{V}(U, B) \bigr) ,
\]

\[
  e_B : \mathrm{Hom}_{V}(U, B) \longrightarrow B
\]

\[
  i_B : B \longrightarrow \mathrm{Hom}_{V}(U, B)
\]

\[
  \mathrm{Hom}_{V}(A, B) \;=\; \mathrm{Hom}_{U}(U \otimes_{V} A, B) ,
\]

\[
  e^{*}_{A} : A \longrightarrow U \otimes_{V} A
  \qquad \text{($V$-linéaire, donné par } a \mapsto 1 \otimes a \text{)}
\]

\[
  i^{+}_{A} : U \otimes_{V} A \longrightarrow A
  \qquad \text{($U$-linéaire surjectif, } u \otimes_{V} a \mapsto ua \text{)}
\]

\[
    H^{i}_{Z}(A_Y) = 0 \quad \text{pour } i \neq 2d
  \]

\[
    H^{i}_{Z}(A_Y) = 0 \quad \text{si } i \neq 2d .
  \]

\[
  \begin{cases}
    R^{q} f_{*}\bigl( f^{*}(F) \bigr) = 0 & \text{pour } 0 < q \leqslant n-1 \\
    R^{0} f_{*}\bigl( f^{*}(G) \bigr) \simeq G &
  \end{cases}
\]

\[
  \cdots \longrightarrow H^{n}(Y, F) \longrightarrow H^{n}(X, F)
  \longrightarrow H^{0}\bigl( Y, R^{n} f_{*}(f^{*}F) \bigr)
\]

\[
  i_{!}\Bigl( p_{*}\bigl( (\mathbf{Z}/n\mathbf{Z})_{Y'} \bigr) \Bigr)
  \;=\; q_{*}\, j_{!}\bigl( (\mathbf{Z}/n\mathbf{Z})_{Y'} \bigr) ,
\]

\[
  R\, g^{!}\bigl( R\, i^{!}(\mu_{\alpha}) \bigr) \longrightarrow
  R\, j^{!}\bigl( A_Y[-2] \bigr)
\]

\begin{align*}
  \Gamma'(F) &= \text{plus grand sous-objet de } F \in \mathrm{Ob}\, C' \\
  \Gamma''(F) &= \text{idem} \in \mathrm{Ob}\, C''
\end{align*}

\[
  \Gamma'/\Gamma''(F) = \Gamma'(F)/\Gamma''(F)
\]

\[
  \Gamma'(F) \longrightarrow \Lambda\,\Gamma'(F)
\]

\[
  \Gamma'x \longrightarrow \Lambda\,\Gamma'x
\]

\[
        R^{0}\bigl( \mathrm{id} / i_{*} i^{!} \bigr) \longrightarrow ST
      \]

\[
        \mathrm{Hom}_{A}(x', x) \xrightarrow{\ \sim\ }
        \mathrm{Hom}_{B}(Tx', Tx) ;
      \]

\[
  f^{*} = (i^{*}, j^{*}) : C \longrightarrow C' \times C'' = D
\]

\[
  \mathrm{Hom}(F, i_{*}G) = \mathrm{Hom}(i^{*}F, G), \qquad
  \mathrm{Hom}(F, j_{*}H) = \mathrm{Hom}(j^{*}F, H)
\]

\[
  f_{*}(G, H) \simeq i_{*}(G) \times j_{*}(H)
\]

\begin{align*}
  \bigl[\ \mathrm{Hom}\bigl( F, f_{*}(G,H) \bigr)
    &\simeq \mathrm{Hom}\bigl( F, i_{*}(G) \bigr) \times
            \mathrm{Hom}(F, j_{*}H) \\
    &\simeq \mathrm{Hom}(i^{*}F, G) \times \mathrm{Hom}(j^{*}F, H) \\
    &\simeq \mathrm{Hom}\bigl( f^{*}F, (G,H) \bigr)\ \bigr]
\end{align*}

\[
  C \xrightarrow{\ \theta\ } K
\]

\begin{align*}
  \mathrm{Hom}(F, j_{*}H) &= \mathrm{Hom}(j^{*}F, H) \\
  \mathrm{Hom}(j_{*}H, F) &\simeq \mathrm{Hom}(H, j^{!}F) \\
  \mathrm{Hom}(F, i_{*}G) &\simeq \mathrm{Hom}(i^{*}F, G) \\
  \mathrm{Hom}(i_{!}G, F) &\simeq \mathrm{Hom}(G, i^{*}F)
\end{align*}

\begin{align*}
  j_{*} j^{!} F &= \Gamma_{Y} F & j_{*} j^{*} F &= F^{Y} \\
  i_{*} i^{*} F &= \Gamma_{U} F & i_{!} i^{*} F &= F^{U}
\end{align*}

\[
  i^{*} j_{*} = 0, \qquad j^{!} i_{*} = 0, \qquad
  j^{*} i_{!} = 0, \qquad j^{!} i_{!} = 0
\]

\[
  j^{*} \longrightarrow \varphi\, i^{*}
\]

\[
        i_{!} i^{*} F \longrightarrow F \longrightarrow j_{*} j^{*} F
        \longrightarrow 0
      \]

\[
        i_{*} i^{*} F \longleftarrow F \longleftarrow j_{*} j^{!} F
        \longleftarrow 0
      \]

\[
        0 \longrightarrow i_{!} i^{*} F \longrightarrow F \longrightarrow
        j_{*} j^{*} F \longrightarrow 0
      \]

\[
  \tau(F) = \mathrm{Ker}\bigl(F \longrightarrow j_{*} j^{*} F\bigr) ,
\]

\[
  \tau(F) \simeq i_{!} i^{*} F
\]

\[
  \mathrm{Hom}(i_{!}G, F) \simeq \mathrm{Hom}(G, i^{*}F) ,
\]

\[
  \mathrm{Hom}(i_{!} i^{*}F', F) \simeq \mathrm{Hom}(i^{*}F', i^{*}F)
\]

\[
  \begin{array}{lll}
    j^{*} i_{!} = 0 & i_{!} = \mathrm{Ker}\, j^{*} & j^{*} = \mathrm{Coker}\, i_{!} \\
    i^{*} j_{*} = 0 & j_{*} = \mathrm{Ker}\, i^{*} & i^{*} = \mathrm{Coker}\, j_{*}
  \end{array}
\]

\[
  \begin{array}{ccccccc}
    0 & \to & N & \to & i_{!} i^{*} F & \to & F \\
    0 & \to & i^{*}N & \to & i^{*} i_{!} i^{*} F & \to & i^{*}F \\
    0 & \to & i_{!} i^{*} N & \to & i_{!} i^{*} i_{!} i^{*} F & \to & i_{!} i^{*} F
  \end{array}
\]

\[
  0 \longrightarrow i_{!} i^{*} F \longrightarrow F \longrightarrow
  j_{*} j^{*} F \longrightarrow 0 .
\]

\[
  j_{*} j^{!} F = \mathrm{Ker}\bigl(F \longrightarrow i_{*} i^{*} F\bigr) .
\]

\[
  0 \to N \to i_{!} i^{*} F \to F , \qquad j^{*}N \to j^{*} i_{!} i^{*} F
  \to j^{*}F .
\]

\[
  j^{*}F = \mathrm{Coker}\bigl(i_{!} i^{*} F \longrightarrow F\bigr) .
\]

\[
  0 \to F_{U} \to F \to F^{Y} \to 0 , \qquad f_{!} f^{*} , \qquad f = i, j
\]

\[
  0 \to \Gamma_{Y} F \to F \to \Gamma_{U} F , \qquad f_{*} f^{!} , \qquad
  f = i, j
\]

\[
  \mathrm{id}_{A} \overset{\alpha}{\longrightarrow} gf , \qquad
  fg \overset{\beta}{\longrightarrow} \mathrm{id}_{B} .
\]

\[
  \varphi \overset{\lambda}{\longrightarrow} \varphi^{2} , \quad\text{i.e.}\quad
  fg \longrightarrow f\,\underbrace{gf}_{\alpha}\,g , \qquad
  \lambda = f * \alpha * g ,
\]

\[
  f(Y) \longrightarrow \Phi^{*}\bigl(f(Y)\bigr)
\]

\[
  f \xrightarrow{\ f * \alpha\ } \Phi \circ f = \varphi f = fgf .
\]

\[
  X \longrightarrow \Phi^{*}(X)
\]

\[
  b \longmapsto \text{classes d'isom.\ de } E_{b}
  = \text{« structures d'espèce } p \text{ sur } b \text{ »}
\]

\[
  \Sigma : B^{\circ} \longrightarrow (\text{ens.\ ordonnés})
\]

\[ F \longrightarrow f^{*}(G) \]

\[ \widetilde{E}^{\circ} \simeq
   \underline{\Gamma}^{\,\text{vert-cart}}\bigl(B,\;
   X \mapsto (\overline{E})^{\circ}_{p(x)/X}\bigr) \]

\[ \underline{\Gamma}\Bigl(B,\; x \mapsto
   \underline{\Gamma}^{\,\mathrm{cart}}\bigl(E_{x},\;
   X \mapsto \underbrace{(\overline{E}_{x})^{\circ}_{/X}}
   _{\textstyle \widetilde{E}^{\circ}_{x} = (\overline{E})^{\circ}_{x}}
   \bigr)\Bigr) \]

\[ \widetilde{E} \simeq \underline{\Gamma}(B, \overline{E}^{\Delta})^{\circ} \]

\[ \underline{\Gamma}\,\varprojlim(B, \overline{E}^{\Delta})^{\circ}
   \simeq \varprojlim(B, \overline{E})^{\Delta}
   \subset \underline{\Gamma}(B, \overline{E})^{\Delta} \]

\begin{align}
\mathrm{Hom}\bigl(X', F^{*}_{c}(Y)\bigr)
  &= \mathrm{Hom}\bigl(i'(X'), F^{*}(i(Y))\bigr) \notag \\
  &= \mathrm{Hom}\bigl(F_{*}\,i'(X'), i(Y)\bigr) \notag \\
  &= \mathrm{Hom}\bigl(\underbrace{i_{*}F_{*}\,i'(X')}
     _{\textstyle F^{c}_{*}(X')}, Y\bigr) \notag
\end{align}

\begin{align}
\mathrm{Hom}\bigl(F^{*}_{c}(Y), X'\bigr)
  &= \mathrm{Hom}\bigl(F^{*}(i(Y)), i'(X')\bigr) \notag \\
  &= \mathrm{Hom}\bigl(i(Y), F_{+}\,i'(X')\bigr) \notag \\
  &= \mathrm{Hom}\bigl(Y, \underbrace{i_{+}F_{+}\,i'(X')}
     _{\textstyle F^{c}_{+}}\bigr) \notag
\end{align}

\[ \dot{F}^{\varepsilon}_{*}(X)(b)
   = \varinjlim_{x \in \mathrm{Ob}\,{}^{b}\!/B} f^{*}(X(x)) \]

\[ \dot{F}^{\varepsilon}_{*}\,\dot{F}^{\varepsilon}_{*}(X) \longrightarrow X \]

\[ \bigcup_{b \in \mathrm{Ob}\,B} b^{c}_{*}(E_{b}). \]

\[ \varprojlim{}^{\circ}(E^{\circ}/B) \xrightarrow{\;d^{\circ}\;}
   \underline{\Gamma}(E^{\circ}/B) \]

\[ \varprojlim(E/B) \xrightarrow{\;j\;}
   \underline{\Gamma}(E^{\circ}/B)^{\circ} \]

\[ \mathrm{Hom}_{\underline{\Gamma}}\bigl(X_{\lambda}, i(Y)\bigr)
   \simeq \mathrm{Hom}_{L}\bigl(i_{*}(X_{\lambda}), Y\bigr), \]

\[ \mathrm{Hom}_{f}(X, Y) \simeq \mathrm{Hom}_{\mathrm{id}_{x}}
   \bigl(X, f^{*}(Y)\bigr) \]

\[ \mathrm{Hom}_{f}(X, Y) \simeq \mathrm{Hom}\,\varphi_{p}(X', Y) \}. \]

\[ F : B \longrightarrow (\mathrm{Cat}) \]

\[ F^{\mathrm{opp}} : B \longrightarrow (\mathrm{Cat}) \]

\begin{align}
F^{\mathrm{opp}}(a) &= F(a)^{\circ} \notag \\
F^{\mathrm{opp}}(f) &= F(f)^{\circ} \notag
\end{align}