Item 1: Interaction Between Precomposition and Postcomposition
For each $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(B,X)$, we have
\begin{align*} [g_{*}\circ f^{*}](\phi ) & = g_{*}(\phi \circ f)\\ & = g\circ (\phi \circ f)\\ & = (g\circ \phi )\circ f\\ & = f^{*}(g\circ \phi )\\ & = [f^{*}\circ g_{*}](\phi ). \end{align*}
Thus $g_{*}\circ f^{*}=f^{*}\circ g_{*}$.
Item 2: Interaction With Composition I
$(g\circ f)_{*}=g_{*}\circ f_{*}$
For each $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(X,A)$, we have
\begin{align*} (g\circ f)_{*}(\phi ) & = (g\circ f)\circ \phi \\ & = g\circ (f\circ \phi )\\ & = g\circ f_{*}(\phi )\\ & = g_{*}(f_{*}(\phi ))\\ & = [g_{*}\circ f_{*}](\phi ). \end{align*}
Thus $(g\circ f)_{*}=g_{*}\circ f_{*}$.
$(g\circ f)^{*}=g^{*}\circ f^{*}$
For each $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(C,X)$, we have
\begin{align*} (g\circ f)^{*}(\phi ) & = \phi \circ (g\circ f)\\ & = (\phi \circ g)\circ f\\ & = (g^{*}(\phi ))\circ f\\ & = f^{*}(g^{*}(\phi ))\\ & = [f^{*}\circ g^{*}](\phi ). \end{align*}
Thus $(g\circ f)^{*}=g^{*}\circ f^{*}$.
Item 3: Interaction With Composition II
It suffices to show the equalities of the maps on $\star \in \mathrm{pt}$. We have
\begin{align*} [g\circ f](\star ) & = g\circ f\\ & = g_{*}(f)\\ & = g_{*}([f](\star ))\\ & = (g_{*}\circ [f])(\star ), \end{align*}
and
\begin{align*} [g\circ f](\star ) & = g\circ f\\ & = f^{*}(g)\\ & = f^{*}([g](\star ))\\ & = (f^{*}\circ [g])(\star ).\end{align*}
Thus $[g\circ f]=g_{*}\circ [f]$ and $[g\circ f]=f^{*}\circ [g]$.
Item 4: Interaction With Composition III
$f^{*}\circ \mathord {\circ ^{\mathcal{C}}_{A,B,C}}=\mathord {\circ ^{\mathcal{C}}_{X,B,C}}\circ (\mathsf{id}\times f^{*})$
For each $(\psi ,\phi )\in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(B,C)\times \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)$, we have
\begin{align*} [f^{*}\circ \mathord {\circ ^{\mathcal{C}}_{A,B,C}}](\psi ,\phi ) & = f^{*}(\psi \circ \phi )\\ & = (\psi \circ \phi )\circ f\\ & = \psi \circ (\phi \circ f)\\ & = \mathord {\circ ^{\mathcal{C}}_{X,B,C}}(\psi ,\phi \circ f)\\ & = \mathord {\circ ^{\mathcal{C}}_{X,B,C}}(\psi ,f^{*}(\phi ))\\ & = [\mathord {\circ ^{\mathcal{C}}_{X,B,C}}\circ (\mathsf{id}\times f^{*})](\psi ,\phi ). \end{align*}
Thus $f^{*}\circ \mathord {\circ ^{\mathcal{C}}_{A,B,C}}=\mathord {\circ ^{\mathcal{C}}_{X,B,C}}\circ (\mathsf{id}\times f^{*})$.
$g_{*}\circ \mathord {\circ ^{\mathcal{C}}_{A,B,C}}=\mathord {\circ ^{\mathcal{C}}_{A,B,D}}\circ (g_{*}\times \mathsf{id})$
For each $(\psi ,\phi )\in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(B,C)\times \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)$, we have
\begin{align*} [g_{*}\circ \mathord {\circ ^{\mathcal{C}}_{A,B,C}}](\psi ,\phi ) & = g_{*}(\psi \circ \phi )\\ & = g\circ (\psi \circ \phi )\\ & = (g\circ \psi )\circ \phi \\ & = \mathord {\circ ^{\mathcal{C}}_{A,B,D}}(g\circ \psi ,\phi )\\ & = \mathord {\circ ^{\mathcal{C}}_{A,B,D}}(g_{*}(\psi ),\phi )\\ & = [\mathord {\circ ^{\mathcal{C}}_{A,B,D}}\circ (g_{*}\times \mathsf{id})](\psi ,\phi ). \end{align*}
Thus $g_{*}\circ \mathord {\circ ^{\mathcal{C}}_{A,B,C}}=\mathord {\circ ^{\mathcal{C}}_{A,B,D}}\circ (g_{*}\times \mathsf{id})$.
Item 5: Interaction With Identities
We have
\begin{align*} \operatorname {\mathrm{id}}^{*}_{A}(\phi ) & = \phi \circ \operatorname {\mathrm{id}}_{A}\\ & = \phi \\ & = \operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)}(\phi )\end{align*}
and
\begin{align*} (\operatorname {\mathrm{id}}_{B})_{*}(\phi ) & = \operatorname {\mathrm{id}}_{B}\circ \phi \\ & = \phi \\ & = \operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)}(\phi )\end{align*}
for each $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)$.
