Precomposition in Functor Categories: Restriction of Group Actions and Restriction of Scalars

Let $[\mathcal{C},\mathcal{D}]$$[\mathcal C,\mathcal D]$ be a functor category, we could consider $[-,\mathcal{D}]:={\mathrm{Hom}}_{\mathrm{Cat}}(-,\mathcal{D})$$[-,\mathcal D]:=\mathrm{Hom}_{\mathrm{Cat}}(-,\mathcal D)$

Let $\mathcal{F}:{\mathcal{C}}^{\prime }\to \mathcal{C}$$\mathcal F:\mathcal C'\to\mathcal C$ be a functor, then we get ${\mathcal{F}}^{\ast }:[\mathcal{C},\mathcal{D}]\to [{\mathcal{C}}^{\prime },\mathcal{D}],{\mathcal{F}}^{\ast }(G)=G\circ F$$\mathcal F^*:[\mathcal C,\mathcal D]\to [\mathcal C',\mathcal D],\mathcal F^*(G)=G\circ F$.

Recall that in Category of G-Set and Category of R-Module: A Uniform Approach via Category Theory

We show that category of $G$$G$-Object and $R$$R$-Module are functor category $[B(G),\mathcal{C}]$$[B(G),\mathcal C]$ and $[B(R):\mathrm{Ab}]$$[B(R):\mathrm{Ab}]$.

Then we could apply it to these particular examples. If you apply it on $[B(R):\mathrm{Ab}]$$[B(R):\mathrm{Ab}]$, then you get the restriction of scalar functor.