I'm not sure what you meant by "both ways", but if you enter the seasonal covariates as survey covariates with each column repeated 5 times, then Presence will use the SST covariates for season 1 for the 1st 5 seasons, since you repeated the covariate 5 times. Presence will only read the 1st 16 columns for colonization and extinction, so do not repeat the columns. Instead, enter each season's SST values in the 1st 16 columns of the survey covariate input matrix. The remaining columns will be ignored, so you don't have to change them. The only other way would be to create 16 site covariates, named SST1, SST2, SST3,...SST16, and use those in the design matrix.
Example design matrix using SST as survey covariates:
gam1 1 SST
gam2 1 SST
gam3 1 SST
:
gam16 1 SST
Example using SST as site covariates:
gam1 1 SST1
gam2 1 SST2
gam3 1 SST3
:
gam16 1 SST16
The first method is a little easier since you can let Presence fill in SST to all rows of the design matrix using the "initial" menu. With the 2nd method, you need to manually enter a different value into each row of the design matrix (unless you use Excel and paste).