Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications.
Let $Y.$ be the semisimplicial set where $Y_1 = \{a, b\}$, $Y_0 = \{u\}$, and $Y_n = \emptyset$ for all $n \geq 2$. We can consider two maps as follows:
$\phi: Y. \rightarrow X.$ defined by $a \mapsto e$, $b \mapsto f$.
$\psi: Y. \rightarrow X.$ defined by $a \mapsto e$, $b \mapsto g$.
Which map from the "figure-eight" to the torus. The first map seems to take the figure-eight to the traditional two non-trivial loops (going inside the hole then back out, and going around the equator). The second map, on the other hand, seems to take the first loop to the same loop on the torus, but the second loop to a combination of the first and second (a kind of diagonal loop composed of one of each of the non-trivial loops from before).
My goal is to compute the kernel and cokernel of $\phi_*$ and $\psi_*$ for each of $H_0, H_1, H_2$, and also to prove that $\psi$ and $\phi$ are not homotopic (which seems intuitively obvious, but for which the proof is escaping me).