Is it true that inverse of a Simplicial Isomorphism is also a Simplicial Isomorphism? Let $\phi$ be a simplicial map and a homeomorphism from $|K_1|$ to $|K_2|$, for any $g \in K_2$, $\phi^{-1} \circ g$ looks like a good simplice that can appear in $K_1$ but I don't think the definition of a Simplicial complex indicated that?
The definition of the related terms I am using:
These definitions are quite nonstandard, and with these definitions it is in fact not true that the inverse of a simplicial isomorphism is simplicial. For instance, let $X=[0,1]$ and let $f_1,f_2:[s]\to X$ be two different homeomorphisms where $[s]$ is a $1$-simplex. Then we can define one simplicial complex $K_1$ in $X$ which consists of $f_1$ and its faces, and another simplicial complex $K_2$ in $X$ which consists of both $f_1$ and $f_2$ and their faces. The identity map $|K_1|\to |K_2|$ is then a simplicial isomorphism, but its inverse is not simplicial.
I would recommend that you ignore the precise definitions in this text, as the author seems to have attempted to make some funny modifications to the standard definitions without carefully thinking through their consequences.