Finite limits from terminal objects and pullbacks

A terminal object and pullbacks in a category give binary products by pulling back the two terminal maps. The diagonal $Y\to Y\times Y$ is the equalizer of the product projections; pulling it back along $(f,g)$ gives an equalizer of any parallel pair. A finite diagram then has a categorical limit as the equalizer of two maps from the product of its object values to the product of its arrow targets. The two maps encode source-image and target projections, so the equalizer enforces every cone equation.