Example 2.1.7.8 (Topologically Enriched Categories). Let $\operatorname{Top}$ denote the category of topological spaces, endowed with the monoidal structure given by the cartesian product (Example 2.1.3.2). We will refer to a $\operatorname{Top}$-enriched category as a topologically enriched category. Note that the functor $F$ of Example 2.1.7.5 is (canonically isomorphic to) the forgetful functor $\operatorname{Top}\rightarrow \operatorname{Set}$. Consequently, if $\operatorname{\mathcal{C}}$ is a topologically enriched category, then the underlying ordinary category $\operatorname{\mathcal{C}}_0$ can be described concretely as follows:
The objects of the ordinary category $\operatorname{\mathcal{C}}_0$ are the objects of the $\operatorname{Top}$-enriched category $\operatorname{\mathcal{C}}$.
Given a pair of objects $X,Y \in \operatorname{\mathcal{C}}_0$, a morphism $f$ from $X$ to $Y$ (in the ordinary category $\operatorname{\mathcal{C}}_0$) is a point of the topological space $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y)$.
Given a pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}_0$, the composition $g \circ f$ is given by the image of $(g,f)$ under the continuous map
\[ c_{Z,Y,X}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Z ). \]
It follows that, for any ordinary category $\operatorname{\mathcal{C}}_0$, promoting $\operatorname{\mathcal{C}}_0$ to a topologically enriched category $\operatorname{\mathcal{C}}$ is equivalent to endowing each of the morphism sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( X, Y )$ with a topology for which the composition maps $\circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( Y, Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( X,Y ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}(X, Z)$ are continuous.