Recall a $k$-space $X$ is a topological space with the property that a subset is open if, and only if its intersection with any compact subspace is open in the subset topology. Let $k$Top be the full subcategory of Top consisting of the $k$-spaces and continuous maps between them.
There is a functor Top $\to$ $k$Top ($k$-ification) changing the topology slightly. We denote $Y_k$ the image of a topological space $Y$ under this functor. It has the property that if $X$ is a $k$-space, $Y$ is any topological space, and $f:X\to Y$ is any function, then $f$ is continuous if, and only if $f$ seen as a map $X\to Y_k$ is continuous.
In light of what we have said above, is it true that a colimit in $k$Top is the same as that colimit taken in Top? Namely, denote by $\iota:$ $k$Top $\to$ Top the inclusion. Let $D$ be a diagram in $k$Top. Is $$\operatorname{colim}^{\textbf{Top}}\iota(D) \cong \iota(\operatorname{colim}^{k\textbf{Top}}D)$$ true?
Yes, since left adjoints preserve colimits.
In more detail, your second paragraph says that $\hom_{\mathbf{Top}}(\iota(X),Y)\cong \hom_{k \mathbf{Top}}(X,Y_k)$. Moreover these bijection are clearly natural (since they are identity at the level of functions). This means that $k$-ification is right adjoint to $\iota$ and hence $\iota$ preserves all colimits.