In the proof of "Joyal's coherence lemma" (lemma 1.6.2 from Cisinski's "Higher categories and homotopical algebra", one can find the following argument :
We have managed to specify three maps $y_i:\Delta^2\to X$, $i=0,1,3$, where $X$ is a simplicial set. Also, each of these maps fits in a commutative triangle that translates the commutativity of a triangle of maps in $X$ i.e. elements of $X_1$. Now comes the part I don't quite understand : the data of $y_0,y_1,y_3$ defines a morphism of $\Lambda_2^3\to X$. Why is that ? At first I assumed I had missed somewhere some property of $\Lambda_k^n$ like an isomorphism $Hom_{sSet}(\Lambda_k^n,X)\simeq ?$ where $?$ would be replaced by something appropriate like $\coprod Hom_{sSet}(...,...)$ or something, but I haven't found any such property in the book nor anywhere else in the literature. I assume this is routine to people familiar and comfortable with simplicial sets, but I am neither of those things so I'm having trouble seeing this. So is there a universal property like I thought there might be ? And how does one go from a triplets of $2$-simplices in $X$ to a map from $\Lambda_2^3$ ?
The horn is defined in Cisinski's book as $\Lambda_k^n:= \cup_{k\in E\subsetneq [n]}\Delta^E$ where $E$ is meant to be a totally ordered set, and $\Delta^E:=N(E)$ the nerve of the poset category, hence $\Delta^E\simeq \Delta^m$ for $m$ the cardinal of $ E$. Given that $Hom$ commutes with limits/colimits I assume that there indeed is such a universal property but I can't quite put my finger on it, partly because I don't really understand the index set the union is over in the definition of the horn.
The required universal property that you are missing is the colimit description of the simplicial horn (when $n\ge2$); $$\Lambda_k^n=\mathrm{coeq}\left(\bigsqcup_{0\le i<j\le n\\i,j\neq k}\Delta^{n-2}\overset{\langle\iota_i\circ\delta_{j-1}\rangle}{\underset{\underset{\langle\iota_j\circ\delta_i\rangle}{\longrightarrow}}{\longrightarrow}}\bigsqcup_{0\le m\le n\\m\neq k}\Delta^{n-1}\right)$$There is a similar presentation of the simplicial boundary for $n\ge2$; $$\partial\Delta^n\cong\mathrm{coeq}\left(\bigsqcup_{0\le i<j\le n}\Delta^{n-2}\overset{\langle\iota_i\circ\delta_{j-1}\rangle}{\underset{\underset{\langle\iota_j\circ\delta_i\rangle}{\longrightarrow}}{\longrightarrow}}\bigsqcup_{0\le m\le n}\Delta^{n-1}\right)$$
In both cases cocone presenting either object as that colimit is given by $\langle\delta_m\rangle$. This is focussed e.g. in Goerss/Jardine’s “Simplicial Homotopy Theory”. It is a good (tedious) exercise to check these.
The upshot is that to specify a map out of the horn $\Lambda^3_2$, the universal property of a coequaliser tells me I only need to find three maps $a,b,c:\Delta^2\to X$ satisfying $c\delta_1=b\delta_2,c\delta_0=a\delta_2,b\delta_0=a\delta_0$ ie we need only present three $2$-simplices such that $d_1c=d_2b, d_0c=d_2a$ and $d_0b=d_0a$. In the textbook I mention, the authors almost always describe morphisms from the horn by such tuples, e.g. they would write $(a,b,-,c)$ here, and this uniquely describes a map. However one must always be careful to ensure the intersection relations hold on the $a,b,c$ ($d,e,f,g,…$) to make this rigorous.