Let $X$ be a simplicial set and let $\wedge_i^2$ be the $i-th$ horn of the simplicial set $\Delta^2$. The Kan condition is the horn filling condition for $i=0,1,2$ and the weak Kan condition is the horn filling condition for $i=1$. In the setting of the diagram
we let $f$, $g$, $h$ be assignments of 1-simplices in $X$ satisfying the obvious compatibility conditions. The Kan condition for $i=0$ says that $g \circ f$ can be defined.
Question: Taking $h$ to be the null degenerate 'identity' assignment, and using the Kan condition for $i=2$, we get that there is a candidate for an inverse of $g$. Is there any sense in which this implies that there is a unique extension of any horn?(In other words, is the difference between weak and regular Kan complexes the uniqueness of the extension of a horn.)
I am already seeing some difficulties with my characterization in the case of the singular set of a topological space. But I just want to find out if the difference between weak and regular Kan complexes can be summarized in terms of something like this.