re this:

> As usual, the left adjoint is 'liberal' or 'generous'. It says "We have to give each German an Italian friend? Okay, let's give them each their own friend!"

Would it be right to say \$$\textrm{Lan}_G(H)\$$ is the **free** \$$\mathcal{C}\$$-database extending the \$$\mathcal{D}\$$-database \$$H\$$?

And that \$$F\circ G\$$ is the **underlying** \$$\mathcal{D}\$$-database of the \$$\mathcal{C}\$$-database \$$F\$$?