Remark 1.3.2.4. The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ of Example 1.3.2.2 induces an equivalence
\[ \{ \text{Categories $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X\} $} \} \xrightarrow {\sim } \{ \text{Monoids} \} . \]
More precisely, there is a pullback diagram of categories
\[ \xymatrix@C =50pt{ \operatorname{Mon}\ar [r]^-{ M \mapsto BM } \ar [d] & \operatorname{Cat}\ar [d]^{ \operatorname{Ob}} \\ \{ \ast \} \ar [r] & \operatorname{Set}, } \]
where $\ast = \{ X \} $ is the set having a single element $X$. Here the upper horizontal functor assigns to each monoid $M$ the category $BM$ of Construction 1.3.2.5, given concretely by
\[ \operatorname{Ob}( BM ) = \{ X \} \quad \quad \operatorname{Hom}_{BM}(X,X) = M. \]