If $\{V_i\}_{i\in I}$ is a family of vector spaces over $F$ with basis $B_i$ for each $V_i$, then there is a vector space $\prod_i V_i$ over $F$, called the direct product of $V_i$'s; its definition involves a certain universal property in terms from projections from it onto $V_i$'s (see this wiki) Since every vector space has a basis, $\prod_i V_i$ has so. Q. Can we obtain basis a of $\prod_i V_i$ from given basis $B_i$ of each $V_i$? I am not too familiar with Category theory; please explain in as elementary fashion as you can, so that this will be also accessible to undergraduates; I want to explain this in my Linear Algebra course to undergraduates, and my aim is to introduce maximum number of advanced concepts of other areas of mathematics from starting point in Linear Algebra. Login To add answer/comment