One way to construct injectives on a presheaf category $[\mathscr C^{\operatorname{op}},\mathbf{Ab}]$ is to consider the forgetful functor
$$i^* \colon \big[\mathscr C^{\operatorname{op}},\mathbf{Ab}\big] \to \big[\mathscr C^{\operatorname{disc,op}},\mathbf{Ab}\big]$$
induced by the inclusion $i \colon \mathscr C^{\operatorname{disc}} \to \mathscr C$ (where $\mathscr C^{\operatorname{disc}}$ is the subcategory with only identity morphisms). If $\mathscr C$ is small, then $i^*$ has left and right adjoints $i_!$ and $i_*$ given by
\begin{align*}
\big(i_! \mathscr F\big)(c) = \bigoplus_{c' \to c} \mathscr F(c'),\\
\big(i_* \mathscr F\big)(c) = \prod_{c \to c'} \mathscr F(c').
\end{align*}
In particular, $i^*$ is an exact left adjoint to $i_*$, so $i_*$ takes injectives to injectives [Stacks, Tag 015N]. But in $[\mathscr C^{\operatorname{disc,op}},\mathbf{Ab}]$ injectives are computed pointwise, so this gives a recipe to construct injectives in $[\mathscr C^{\operatorname{op}},\mathbf{Ab}]$.
See for example [Stacks, Tag 01DJ] for a brief discussion, or [SGA IV$_1$, Exp. I, Prop. 5.1] for a more general discussion of adjoints (but without the mention of injectives).
In general the colimit for $i_!$ is taken over the opposite of the comma category $(i \downarrow c)$ (whose objects are $(i(c') \to c)$), which in this case is just a discrete category since $\mathscr C^{\operatorname{disc}}$ is, so we get a direct sum; similarly for $i_*$.
References.
[SGA IV$_1$] M. Artin, A. Grothendieck, J.-L. Verdier, Séminaire de géométrie algébrique du Bois-Marie 1963–1964. Théorie de topos et cohomologie étale des schémas (SGA 4), 1: Théorie des topos. Lecture Notes in Mathematics 269. Springer-Verlag (1972). ZBL0234.00007.
[Stacks] A.J de Jong et al, The stacks project.
