"Since faithful functors are injective on $\text{hom}$-sets, we usually assume...that $\operatorname{hom}_{\mathbf{A}}(A, B)$ is a **subset** of $\operatorname{hom}_{\mathbf{X}}(UA, UB)$ for each pair $(A, B)$ of $\mathbf{A}$-objects. This familiar convention allows one to express the property that "for $\mathbf{A}$-objects $A$ and and an $\mathbf{X}$-morphism $f:UA \to UB$ there exists a (necessarily unique) $\mathbf{A}$-morphism $A \to B$ with $\require{AMScd}U(A \to B) = \begin{CD}
UA @>{f}>> UB\end{CD}$" much more succinctly, by stating