Example 6.15.5. The lemma will be applied often to the following situation. Suppose that we have a diagram

$\xymatrix{ A \ar[r] & B \ar[d] \\ C \ar[r] & D }$

in $\mathcal{C}$. Suppose $C \to D$ is injective on underlying sets, and suppose that the composition $A \to B \to D$ has image on underlying sets in the image of $C \to D$. Then we get a commutative diagram

$\xymatrix{ A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D }$

in $\mathcal{C}$.

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