Technically, the stigma is the female sex part of a non-orchidacious flower, where the anthers are the male parts that hold the pollen.
Orchids are orchids because their male and female sex parts are fused into a column. That would be the "nose" of the cattleya. The anther cap sits at the end of the column and protects the pollinia, which contains the pollen. The pollen then gets applied to the underside of the column, which is very sticky.
Technically, the column should swell and wrap itself around the pollen. But when I crossed my Catts, I simply took the whole pollinia and stuck it onto the column with the pollen-side in contact. The column never enclosed the pollinia.
If the other flowers and stems start to fade and the stem leading to the pollinated flower stays green and healthy (the flower itself will turn brown and die off), then you're probably in business!