Is every kernel pair a pullback?

Question

The limit of a functor $F\colon\mathcal{D}\to\mathcal{C}$ is defined as a pair $(L,(p_D)_{D\in\mathcal{D}})$, where for each $D$ in $\mathcal{D}$ there is a morphism $p_D\colon L\to F(D)$, such that some properties hold... So, a kernel pair of a morphism $f\colon A\to B$ has the two projections always equal (to $p_A$), which is absurd. What am I misunderstanding?

Answer 1

A kernel pair is indeed a limit, as it can be defined as the pullback $$\require{AMScd} \begin{CD}A\times_B A@>{\pi_2}>> A \\ @V{\pi_1}VV @VV{f}V \\ A @>>{f}> B. \end{CD}$$

The reason you are allowed to have two distinct projections to $A$ is simply that in the diagram whose limit we're taking, the object $A$ appears twice.