ct.category theory - What theorem constructs an initial object for this category? (Formerly "Integrability by abstract nonsense")

The category you described can be written as a lax limit of a diagram in the 2-category of categories. The diagram in question consists of accessible categories and accessible functors, so its limit is again accessible by a theorem of Makkai and Paré. It is obvious from the construction that the category has all small limits (they are computed as in the category of Banach spaces and nonexpanding maps), so it is in fact locally presentable. This means that it has all colimits and in particular an initial object.

Here are some more details and references. Let $mathbf{Ban}_1$ be the category of Banach spaces and nonexpanding maps. This category is locally $aleph_1$-presentable (see e.g. Borceux, Handbook of Categorical algebra, Volume II, 5.2.2.e). Let

$F colon mathbf{Ban}_1 rightarrow mathbf{Ban}_1$

be the functor which sends a Banach space $X$ to $mathbb{R}+Xoplus X$, where + stands for the coproduct. A morphism f of Banach spaces gets sent to $mathrm{id}_{mathbb{R}}+foplus f$. Let $U colon mathbf{Ban}_1 rightarrow mathbf{Set}$ be the functor which sends a Banach space to its underlying set. Since $aleph_1$-filtered colimits are computed as in the category of sets (see e.g. Borceux, Handbook of Categorical algebra, Volume II, 5.2.2.e) we know that the composite UF preserves $aleph_1$-filtered colimits. By the open mapping theorem it follows that $F$ preserves $aleph_1$-filtered colimits.

By the theorem of Makkai and Paré (see e.g. Adámek, Rosický, Locally Presentable and accessible categories, Theorem 2.77), the inserter $mathcal{C}$ of $F$ and the identity functor on $mathbf{Ban}_1$ is again an accessible category. The objects of $mathcal{C}$ are triples $(X,xi,u)$ where $xicolon Xoplus X rightarrow X$ and $ucolon mathbb{R} rightarrow X$ are nonexpanding (i.e., u corresponds to an element of $X$ of norm less than or equal one). The morphisms are morphisms of Banach spaces which are compatible with $u$ and $xi$. Thus $mathcal{C}$ is almost the category we are interested in; the only thing that's missing is the requirement that $xi(u,u)=u$.

Let

$G colon mathcal{C} rightarrow mathbf{Ban_1}$

be the functor which sends every object to $mathbb{R}$ and every morphism to $mathrm{id}_{mathbb{R}}$. This is clearly a functor which preserves $aleph_1$-filtered colimits. Let

$H colon mathcal{C} rightarrow mathbf{Ban_1}$

be the functor which sends $(X,xi,u)$ to $X$ and a morphism to itself; this is again a functor which preserves $aleph_1$-filtered colimits. There are two natural transformations $alpha,beta colon G Rightarrow H$, whose component at $(X,xi,u)$ is given by $u$ and $xi(u,u)$ respectively. The equifier of $G$ and $H$ is precisely the category we are looking for, and from our construction we can conclude that it is accessible.

The obvious forgetful functor to Banach spaces creates limits, so this category is complete. Thus it is locally presentable, and therefore also cocomplete. In particular, it follows that our category has an initial object.