I am considering the $(2,1)$-category of Graphs as defined by Tien Chich and Laura Scull in "A homotopy category of graphs". Its objects are single-edged non-directed graphs, and the $1$-morphisms are the usual graph morphisms. In this $1$-category, the graph with a single vertex with self loop is the terminal object.
Consider now the notion of $\times$-homotopy between graph maps $f,g:G\to H$. Let $I_n^l$ be the $n$-th looped interval, namely the graph with $n+1$ vertices $\{0,1,\ldots,n\}$ and edges $i\sim i+1$ for all $0\leq i<n$. A length $n$ $\times$-homotopy betweeen $f$ and $g$ is a map $\Lambda:G\times I_n^l\to H$ for some $0\leq n<\infty$. Those homotopies will be the $2$-cells in the $2$-category of graphs, up to an appropriate notion of equivalence that I describe now.
In definition 3.12, they say that two $\times$-homotopies $\alpha,\alpha'$ from $f$ to $g$ are themselves homotopic if they are homotopic rel endpoints viewed as looped walks in $(H^G)^{I_n^l}$. We don't need to understand those notions, but only to realize that the latter object, which denotes the internal hom of maps $I_n^l\to H^G$, indicates that $\alpha$ and $\alpha'$ are both $\times$-homotopies of length $n$.
They then go on to show (Thm 3.18) that if we take the $2$-morphisms between $f$ and $f'$ to be homotopy rel endpoints classes $[\alpha]$ of $\times$-homotopies $\alpha:I^l_n\to H^G$ such that $f\simeq f'$, this forms a $2$-category $Gph$.
My question is: is the terminal object in the $1$-category of graphs (as described above) a terminal object in the $2$-category $Gph$ ?
My confusion comes from the fact that they don't seem to identify two homotopies $\alpha$ and $\alpha'$ if they don't have the same length. However, if we consider the $1$-category $Gph(1,1)$ where $1$ is the terminal object described above, then this category has a unique object, namely $!_1:1\to1$, but its set of morphisms is $\{[\alpha_n]\vert \alpha_n:I_n^l\to 1^1, n\in\mathbb{N}\}\cong\mathbb{N}$. Hence $Gph(1,1)(!_1,!_1)$ is not the terminal category, and so $1$ would not be terminal in the $2$-category $Gph$.
It seems that the above homotopies $\alpha_n:I_n^l\times 1\to 1$ should really be identified for various $n$, but the definitions seems not to suggest that.
