  In this paper, we provide an accessible introduction to the theory of locally
convex supermanifolds in the categorical approach. In this setting, a
supermanifold is a functor $\mathcal{M}\colon\mathbf{Gr}\to\mathbf{Man}$ from
the category of Grassmann algebras to the category of locally convex manifolds
that has certain local models, forming something akin to an atlas. We give a
mostly self-contained, concrete definition of supermanifolds along these lines,
closing several gaps in the literature on the way. If $\Lambda_n\in\mathbf{Gr}$
is the Grassmann algebra with $n$ generators, we show that
$\mathcal{M}_{\Lambda_n}$ has the structure of a so called multilinear bundle
over the base manifold $\mathcal{M}_\mathbb{R}$. We use this fact to show that
the projective limit $\varprojlim_n\mathcal{M}_{\Lambda_n}$ exists in the
category of manifolds. In fact, this gives us a faithful functor
$\varprojlim\colon\mathbf{SMan}\to\mathbf{Man}$ from the category of
supermanifolds to the category of manifolds. This functor respects products,
commutes with the respective tangent functor and retains the respective
Hausdorff property. In this way, supermanifolds can be seen as a particular
kind of infinite-dimensional fiber bundles.
