By saying "only", I mean that no words are involved except "commutative". Just draw a commutative diagram, and make some arrows dotted lines to mean "the other arrows already exist, and this is what the theorem demonstrates to exist".
To prove a theorem like this, we first draw all the conditions as solid arrows, and try to prove that a new arrow exists. We can use other theorems in this style by stacking them up to show the existence of other arrows in the diagram. Then we might get to the arrow we want, everything like doing a high school geometry problem.
However, It seems difficult to rewrite the theorems in linear algebra or abstract algebra, because it involves too many statements that are not shown in the commutative diagram, like injectivity, or the kernel being connected in another kernel, or some maps being "canonical". For example,
Suppose $\varphi:E\to F$ and $\psi: E\to G$ are linear mappings such that $\ker\varphi\subset\ker \psi$, then exists $\chi:F\to G$ such that $\chi\circ\varphi = \psi$.
The last statement can certainly be a commutative diagram (I can't draw it here), and $\chi$ is a dotted line. but where is the crucial $\ker\varphi\subset\ker \psi$ condition? Therefore, I can't figure it out.
Is there some ways that make this idea possible?