\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathfrak{P} & \mathcal{D} \arrow[rd] & & \\
\mathcal{L} = \mathcal{D} \cap N \arrow[ru] \arrow[rd, "?"'] & & \mathcal{D} + N = E_{0} \arrow[r] & E \\
& N \arrow[ru] & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\underline{K}(X) \arrow[r, "\text{inj.}"] \arrow[d, "\widetilde{c}_{X}"'] & \underline{K}(X \times \mathbb{P}^{r}) \arrow[d, "\widetilde{c}_{X \times \mathbb{P}^{r}}"] \\
\widetilde{A}(X) \arrow[r] & \widetilde{A}(X \times \mathbb{P}^{r})
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=small]
K_{\Phi}(X) \arrow[r] \arrow[d, "c_{\Phi}"'] & K(X) \arrow[d, "c"] \\
A_{\Phi}(X) \arrow[r] & A(X)
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
K^{\bullet}(i_{n}) \arrow[r, "u"] \arrow[d, "\alpha"'] & \varinjlim_{n} K^{\bullet}(i_{n}) \arrow[r] \arrow[d, "\beta"] & K^{\bullet}_{X}(X') \arrow[r] \arrow[d, "\gamma"] & K^{\bullet}(X') \arrow[d, "\delta"] \\
K_{\bullet}(X_{n}) \arrow[r, "u'"'] & \varinjlim_{n} K_{\bullet}(X_{n}) \arrow[r] & K^{X}_{\bullet}(X') \arrow[r, "w'"'] & K_{\bullet}(X')
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=small]
K(X) \arrow[r, "f_{!}"] \arrow[d, "\widetilde{c}_{X}"'] & K(Y) \arrow[d, "\widetilde{c}_{Y}"] \\
\widetilde{A}(X) \arrow[d, "h_{X}"'] & \widetilde{A}(Y) \arrow[d, "h_{Y}"] \\
A(X) \otimes \mathbb{Q} \arrow[d, "\tau_{h}(X)"'] & A(Y) \otimes \mathbb{Q} \arrow[d, "\tau_{h}(Y)"] \\
A(X) \otimes \mathbb{Q} \arrow[r, "f_{*}"] & A(Y) \otimes \mathbb{Q}
\end{tikzcd}