A functor $F:\mathcal{C}\to \mathcal{D}$, from an essentially small category to a cocomplete category induces a realisation-nerve adjunction between the categories $\mathbf{Fun}(\mathcal{C}^{op},\mathbf{Set})$ and $\mathcal{D}$. I vaguely recall reading that "the nerve functor deserves the name singular functor only if it is fully faithful". For the standard cosimplicial topological space $\Delta^\bullet_{top}:\mathbf{\Delta}\to \mathbf{Top}$, one retrieves the geometric realisation-singular simplicial adjunction. It is easy to see that singular simplicial functor is faithful. But, I do not see why it is full, if it is.

The singular simplicial functor is fully faithful iff the left Kan extension $\mathbf{Lan}_{\Delta^\bullet_{top}}\Delta^\bullet_{top}$ is point-wise and is isomorphic to the identity functor. Hence, I am 'almost-convinced' that the singular simplicial functor is not full, and an alternative question would be:

what is an example of a topological space $X$, for which $\mathbf{Lan}_{\Delta^\bullet_{top}}\Delta^\bullet_{top}(X)\ncong X$?

PS I assumed earlier that my question should be well-known, and I have asked this question on math.stackexchange.com, but it does not seem to be the case.