The transverse singular complex

Let $M$ be a smooth manifold without boundary and let $\mathcal{T}$ be a countable collection of manifolds with corners, each equipped with a smooth map to $M$. We show that the singular simplicial set $\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(M)$ of $M$ deformation retracts onto the simplicial subset ${\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}}^{\mathcal{T}}\text{\hspace{0.17em}⁣}(M)$ of smooth singular simplices that are transverse to every element of $\mathcal{T}$.