Let $R_i$ be a family of associative algebras s.t. $\phi_{ij}:R_i\to R_j$ is unital associative algebra homomorphism where I demand 1 being sent to 1 and I am assuming this is directed limit.
$\textbf{Q:}$ Is $lim R_i$ unital algebra? It seems that I can form limit in the set category first and endow associative algebra structure. In other words, category of associative algebras is co-complete?
$\textbf{Q':}$ Consider $Q$ quiver of infinite size.(i.e. It has infinite vertices . I am going to assume number of arrows between vertices is finite here.) Now approximate $Q$ by subquivers $Q_i$ with $Q_i$ forming ascending filtration of $Q$ where $i$ is the number of vertices. Consider path algebra $KQ_i$. Clearly $K$ is a functor from quiver category to associative algebra(not necessarily unital). So $KQ$ is non unital but each $KQ_i$ has unit. So $K$ does not preserve the direct limit. Am I wrong on this part? The possible place is $K$ being a functor from quiver category to associative algebras rather a functor from finite quiver category to associative algebras.