One way to approach the concepts of "elements" (or "its") and "distinctions" (or "dits") is to start with the category-theoretic duality between subsets and quotient sets (= partitions = equivalence relations): "The dual notion (obtained by reversing the arrows) of `part' [subobject] is the notion of partition." (Lawvere & Roseburgh, p. 85). That motivates the two dual forms of mathematical logic: the Boolean logic of subsets and the logic of partitions (Ellerman, David. 2014. “An Introduction to Partition Logic.” Logic Journal of the IGPL 22 (1): 94–125. https://doi.org/10.1093/jigpal/jzt036) . If partitions are dual to subsets, then what is the dual concept that corresponds to the notion of elements of a subset? The notion dual to the elements of a subset is the notion of the distinctions or dits of a partition (pairs of elements in distinct blocks of the partition).
The duality between elements ("its") of a subset and distinctions ("dits") of a partition already appears in the very notion of a function between sets in the category $Set$. The concepts of elements and distinctions provide the natural notions to specify the binary relations, i.e., subsets $R\subseteq X\times Y$, that define functions $f:X\rightarrow Y$.
A binary relation $R\subseteq X\times Y$ transmits elements if for each element $x\in X$, there is an ordered pair $\left( x,y\right) \in R$ for some $y\in Y$.
A binary relation $R\subseteq X\times Y$ reflects elements if for each element $y\in Y$, there is an ordered pair $\left( x,y\right) \in R$ for some $x\in X$.
A binary relation $R\subseteq X\times Y$ transmits distinctions if for any pairs $\left( x,y\right) $ and $\left( x^{\prime},y^{\prime }\right) $ in $R$, if $x\not =x^{\prime}$, then $y\not =y^{\prime}$.
A binary relation $R\subseteq X\times Y$ reflects distinctions if for any pairs $\left( x,y\right) $ and $\left( x^{\prime},y^{\prime}\right) $ in $R$, if $y\not =y^{\prime}$, then $x\not =x^{\prime}$.
The dual role of elements and distinctions can be seen if we translate the usual characterization of the binary relations that define functions into the elements-and-distinctions language. In the usual treatment, a binary relation $R\subseteq X\times Y$ defines a \textit{function} $X\rightarrow Y$ if it is defined everywhere on $X$ and is single-valued. But "being defined everywhere" is the same as transmitting (or "preserving") elements, and being single-valued is the same as reflecting distinctions so the more natural definition is:
a binary relation $R$ is a function if it transmits elements and reflects distinctions.
What about the other two special types of relations, i.e., those which transmit (or preserve) distinctions or reflect elements? The two important special types of functions are the injections and surjections, and they are defined by the other two notions:
a function is injective if it transmits distinctions, and
a function is surjective if it reflects elements.
The origin of the 'reverse-the-arrows' duality is in the definition of the concrete morphisms in $Set^{op}$, i.e., cofunctions, which is interchange the role of elements and distinctions in the definition of a function. That is, a cofunction is a binary relation the transmits distinctions and reflects elements. Then the notion of functions and cofunctions is abstracted in abstract categories to give the reverse-the-arrows duality.
It all traces back to the two dual logics of subsets and partitions, a duality that at the concrete or granular level is the elements & distinction or its & dits duality--since reversing the roles of elements and distinctions in the ur-category of $Sets$ to go to $Sets^{op}$ just reverses the arrows.