Consider the abelian (Grothendieck) category $\mathcal{C} := \mathrm{Fun}(\{0<1\},\mathrm{Ab}) = \mathrm{Mor}(\mathrm{Ab})$. Objects are morphisms $(A \to B)$ of abelian groups, morphisms are commutative diagrams. Equivalently, this is the category of abelian sheaves on the Sierpinski space $\{0,1\}$ with the non-trivial open subset $\{1\}$.

Question. How do injective objects in $\mathcal{C}$ look like?

Since injective sheaves are stable under restriction (use extension by zero), clearly $(A \to B)$ injective implies that $A$ is injective. But is this sufficient (probably not)? When $A,B$ are injective, is the same true for $(A \to B)$?

Consider the abelian (Grothendieck) category $\mathcal{C} := \mathrm{Fun}(\{0<1\},\mathrm{Ab}) = \mathrm{Mor}(\mathrm{Ab})$. Objects are morphisms $(A \to B)$ of abelian groups, morphisms are commutative diagrams. Equivalently, this is the category of abelian sheaves on the Sierpinski space.

Question. How do injective objects in $\mathcal{C}$ look like?

Since injective sheaves are stable under restriction (use extension by zero), clearly $(A \to B)$ injective implies that $A$ is injective. But is this sufficient (probably not)? When $A,B$ are injective, is the same true for $(A \to B)$?