What follows is preamble to motivate the question; for users who prefer to just see the question, please scroll down to the end of the question.

Let $\mathcal{C}$ be a category, using the one hom-class definition. For each object $X\in{\bf Ob}_\mathcal{C}$, we define $\overline X$ to be the set of all arrows in $\mathcal{C}$ with codomain $X$; that is, $$\overline X = \{f\in{\bf Hom}_\mathcal{C}:cod(f)=X\}.$$ For each arrow $f:X\to Y\in{\bf Hom}_\mathcal{C}$, we define a function $$f\circ:\overline X\to\overline Y$$ $$g\mapsto f\circ g.$$ More generally, we will say that a function $g:\overline X\to\overline Y$ respects domains iff $g$ commutes with the domain selection function in $\mathcal{C}$, so for all $f\in\overline X$ $$dom(g(f))=dom(f).$$ We will say that $g$ is precomposition linear iff for all objects $f\in\overline X$ and all objects $h\in\overline{dom(f)}$ we have that $$g(f\circ h)=g(f)\circ h.$$ Note that postcomposition functions trivially satisfy both conditions. This is essentially just a repackaging of the data of hom-presheaves and natural transformations between them 'one dimension lower', where we encode the extra dimensional information as axioms on the collapsed data. In this case, the separation into hom-sets that hom-presheaves usually provide is taken care of by requiring functions between collections of generalized global elements to respect domains, and naturality is taken care of by precomposition linearity.

$\DeclareMathOperator\dom{dom}$What follows is preamble to motivate the question; for users who prefer to just see the question, please scroll down to the end of the question.

Let $\mathcal{C}$ be a category, using the one hom-class definition. For each object $X\in{\bf Ob}_\mathcal{C}$, we define $\overline X$ to be the set of all arrows in $\mathcal{C}$ with codomain $X$; that is, $$\overline X = \{f\in{\bf Hom}_\mathcal{C}:\operatorname{cod}(f)=X\}.$$ For each arrow $f:X\to Y\in{\bf Hom}_\mathcal{C}$, we define a function $$f\circ:\overline X\to\overline Y$$ $$g\mapsto f\circ g.$$ More generally, we will say that a function $g:\overline X\to\overline Y$ respects domains iff $g$ commutes with the domain selection function in $\mathcal{C}$, so for all $f\in\overline X$ $$\dom(g(f))=\dom(f).$$ We will say that $g$ is precomposition linear iff for all objects $f\in\overline X$ and all objects $h\in\overline{\dom(f)}$ we have that $$g(f\circ h)=g(f)\circ h.$$ Note that postcomposition functions trivially satisfy both conditions. This is essentially just a repackaging of the data of hom-presheaves and natural transformations between them 'one dimension lower', where we encode the extra dimensional information as axioms on the collapsed data. In this case, the separation into hom-sets that hom-presheaves usually provide is taken care of by requiring functions between collections of generalized global elements to respect domains, and naturality is taken care of by precomposition linearity.

This is all ultimately justified by Yoneda, since transformations between hom-presheaves come uniquely from arrows in the underlying category. For example, consider exponential objects -- for objects $X,Y\in{\bf Ob}_\mathcal{C}$, we can define an exponential object without finite products using this method in the same way that we can using presheaves. That is, an exponential object for $X$ and $Y$ consists of an object $Z$ together with a function $$\overline\epsilon:\overline Z\times\overline Y\to \overline X$$ which respects domains and is precomposition linear such that for any other object $A$ together with a function $$\overline f:\overline A\times\overline Y\to\overline X$$ which respects domains and is precomposition linear, there exists a unique function $$\overline{\tilde{f}}:\overline A\to\overline Z$$ such that

This is all ultimately justified by Yoneda, since transformations between hom-presheaves come uniquely from arrows in the underlying category. For example, consider exponential objects -- for objects $X,Y\in{\bf Ob}_\mathcal{C}$, we can define an exponential object without finite products using this method in the same way that we can using presheaves. That is, an exponential object for $X$ and $Y$ consists of an object $Z$ together with a function $$\overline\epsilon:\overline Z\times\overline Y\to \overline X$$ which respects domains and is precomposition linear such that for any other object $A$ together with a function $$\overline f:\overline A\times\overline Y\to\overline X$$ which respects domains and is precomposition linear, there exists a unique function $$\overline{\tilde{f}}:\overline A\to\overline Z$$ which respects domains and is precomposition linear such that

What follows is preamble to motivate the question; for users who prefer to just see the question, please scroll down to the first highlighted section below.

What follows is preamble to motivate the question; for users who prefer to just see the question, please scroll down to the end of the question.