Let $\Delta$ denote the "ordinal number category" whose objects are categories $\bf{n}$ where $\mathbf{n}=0\to1\to2\to\cdots\to n$ where identity morphisms are suppressed.
A morphism in $\Delta$ is $\theta:\mathbf{n}\to\mathbf{m}$ is just a functor.
A simplicial set is a functor $X:\Delta^{\text{op}}\to \text{Sets}$. Let us denote the category of simplicial sets by $\mathbf{S}$, whose objects are simplicial sets, and whose morphisms are natural transformations of functors.
The standard simplicial n-simplex is $\Delta^n=\hom_\Delta(-,\mathbf{n})$.
From Yoneda lemma we have $\hom_\mathbf{S}(\Delta^n,Y)=\hom_{\mathbf{S}}(\hom_{\Delta}(-,\mathbf{n}),Y)=\hom_\mathbf{S}(h_\mathbf{n},Y)\cong Y(\mathbf{n})=Y_n$.
Goerss - Simplicial homotopy theory - page 6:
Let $\iota_n=1_\mathbf{n}\in\hom_{\Delta}(\mathbf{n},\mathbf{n})$. Then the bijection [of Yoneda] is given by associating the simplex $\varphi(\iota_n)\in Y_n$ to each simplicial map $\varphi:\Delta^n\to Y$.
I don't understand this statement. $\varphi$ is a natural transformation of functors, and $1_n$ is an element of the component $\Delta^n(\mathbf{n})$. So although there is a map $\varphi(\mathbf{n}):\Delta^n(\mathbf{n})\to Y_n$, such that we can consider $\varphi(\mathcal{n})(\iota_n)\in Y_n$, I don't see what $\varphi(\iota_n)$ should mean.