The following are taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes $\textit{elementary categories, elementary toposes}$ by McLarty
$\quad$$\textbf{Definition 1: }$ Given $f_1:A_1\rightarrow A$ and $f_2:A_2\rightarrow A$, we say that the commutative diagram
is a $\textbf{pullback}$ (also $(g_1, g_2)$ is a $\textbf{pullback}$ of $(f_1,f_2),$ $g_1$ is a $\textbf{pullback of }$ $f_1$ $\textbf{along}$ $f_2$, etc), if it has the property that any commutative diagram $f_2\circ g'_1=f_1\circ g'_2$
![[Image]](https://i.stack.imgur.com/Zban2.jpg)
can be completed by a unique $\psi$ as show above. The dual construction is called a $\textbf{pushout}.$ [Image]
$\quad$$\textbf{Definition 2: }$ For any arrow $f:A\rightarrow B$ the projection arrows in a pullback of $f$ along itself are called a $\textit{kernel pair}$ for $f$
$\textbf{Questions:}$
What I would like to know is that what does it mean in the definition of a pullback, the phrase "$g_1$ $\text{ is a }$ $\textbf{pullback of }$ $f_1$ $\textbf{along}$ $f_2.$ Similarly in the definition for kernel pair, in terms of pullback square commutative diagram, what does it means to say that a projection function $f$ in a "pullback along itself". I am not sure if I am suppose to visualize some sort of motion along the direction of the arrows in these commutative diagrams. Thank you in advance