  We define the path coalgebra and Gabriel quiver constructions as functors
between the category of $k$-quivers and the category of pointed $k$-coalgebras,
for $k$ a field. We define a congruence relation on the coalgebra side, show
that the functors above respect this relation, and prove that the induced
Gabriel $k$-quiver functor is left adjoint to the corresponding path coalgebra
functor. We dualize, obtaining adjoint pairs of functors (contravariant and
covariant) for pseudocompact algebras. Using these tools we describe precisely
to what extent presentations of coalgebras and algebras in terms of path
objects are unique, giving an application to homogeneous algebras.
