Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is a very general structure for all constructs:
A simple $F$-construct is a set and a binary relation $(X,\mathcal R)$ such that $\mathcal R\subset F(X)\times X$. The functor $F$ characterize the primary structure that determines the conditions on the morphisms and there could be secondary axioms on $\mathcal R$ separating categories from each other. More general constructs can be obtained by combining simple constructs.
For example, if $F(X)=X\times X$, then
$(X,\mathcal R)$ could be anything from categories
$(X=\text{Mor}(\underline C)$ with
$((\alpha,\beta),\gamma)\in\mathcal R \iff \beta\alpha=\gamma)$ to groups when $\mathcal R$ is an associative function etc.
Non algebraic examples are topological spaces
$F(X)=\mathcal P(X)$ (power set) with $\mathcal (A,x)\in \mathcal R\iff x\in \bar A$
What I'm looking for is possible counterexamples, known and newly found constructs, that doesn't seems to fit in to the pattern.
As Tobias Kildetoft pointed out in a comment a relation is a very general approach $-$ and that is intentional. But $F$-constructs is about a normal form for all constructs as defined above and given in this normal form there is a general rule that determine the condition for a function to be a morphism.
A morphism is a function $f:X\rightarrow Y$ satisfying the relation:
$(1)\quad\; (\phi,\psi)\in F(f)\Rightarrow\left[(\phi,x)\in\mathcal R^F_X\Rightarrow(\psi,f(x))\in\mathcal R^F_Y\right]$.
The meaning with the construction of the relation is demonstrated by the diagram: $\require{cancel}$ $\require{AMScd}$ \begin{CD} F(X) @>F(f)>> F(Y)\\ @V \mathcal R_X^F VV\xcancel{\#} @VV\mathcal R_Y^F V\\ X @>>f> Y \end{CD}
The upper row gives a condition on $f$ on the lower row, together with the relations, and the connection is given by $(1)$. (The diagram commutes if the $\mathcal R$:s are functions and $F(f)$ is a function or a anti-function.
Of course, if someone find a counterexample of this condition, I would be interested to know that too.
Examples of how $(1)$ works could be find here.
I really don't understand the resistance against the question. The method works, and it is not easy to find adequate counterexamples.