Remark 11.10.3.4. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{D}}$. Then the image $U(f)$ can be identified with a functor from the partially ordered set $[1] = \{ 0 < 1 \} $ to the category $\operatorname{\mathcal{C}}$ (carrying $0$ to the object $U(X)$, and $1$ to the object $U(Y)$). Form a pullback diagram of categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}' \ar [r] \ar [d]^-{U'} & \operatorname{\mathcal{D}}\ar [d]^-{U} \\ {[1]} \ar [r]^-{U(f)} & \operatorname{\mathcal{C}}. } \]
By construction, $f$ can be lifted uniquely to a morphism $f': X' \rightarrow Y'$ in the category $\operatorname{\mathcal{D}}'$ satisfying $U'(X') = 0$ and $U'(Y') = 1$. Then: