Remark 7.5.2.5. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cat}$ be a functor of ordinary categories. Then the category $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ can be described more concretely as follows:
- $(1)$
An $M \in \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is a rule which assigns to each object $C \in \operatorname{\mathcal{C}}$ an object $M(C) \in \mathscr {F})$, and to each morphism $u: C \rightarrow D$ in $\operatorname{\mathcal{C}}$ an isomorphism $M(u): \mathscr {F}(u)( M(C) ) \xrightarrow {\sim } M(D)$, subject to the following constraints:
For each object $C \in \operatorname{\mathcal{C}}$, $M( \operatorname{id}_{C} )$ is the identity morphism from $M(C)$ to itself.
For every pair of composable morphisms $u: C \rightarrow D$ and $v: D \rightarrow E$ of $\operatorname{\mathcal{C}}$, we have $M( v \circ u ) = M(v) \circ \mathscr {F}(v)( M(u) )$.
- $(2)$
If $M$ and $N$ are objects of $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$, then a morphism $\alpha : M \rightarrow N$ in $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is a rule which assigns to each object $C \in \operatorname{\mathcal{C}}$ a morphism $\alpha _{C}: M(C) \rightarrow N(C)$ in the category $\mathscr {F}(C)$, subject to the following constraint:
For every morphism $u: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the diagram
\[ \xymatrix@C =50pt@R=50pt{ \mathscr {F}(u)( M(C) ) \ar [d]^{ \mathscr {F}(u)( \alpha _ C )} \ar [r]^-{ M(u) }_{\sim } & M(D) \ar [d]^{ \alpha _{D} } \\ \mathscr {F}(u)( N(C) ) \ar [r]^-{ N(u) }_{\sim } & N(D) } \]commutes (in the category $\mathscr {F}(D)$).