In 1874 the Earl of Shaftesbury started a charity to provide shelters for Cabmen, where they could get a hot meal while working. They were not allowed to leave their cabs, so they had to be situated by (or in) the road. 13 still exist (of 61 built between 1875 & 1914) and all are Grade II listed buildings.
Behind it on the left is The Playhouse Theatre, with Charing Cross Station on the right.
1888 puts it in the William Ewart Gladstone era; but is it he? If not, who then? There are apocryphal stories of notable Conservatives confusing the NLC for a public convenience when on their way to the House. Perhaps this is one of 'them' leaving.